# calculus

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A triangle ABC has sides of length AB = 3 inches and AC = 4 inches. Let t denote the angle at vertex A, and let s denote the length of the remaining side BC. If t increases at the rate of 2 radians per second, at what rate (in inches per second) does s change at the instant when t is a right angle? [Hint: You will need to use the Law of Cosines.]

• calculus -

s^2 = 3^2 + 4^2 - 2(3)(4)cos T
2s ds/dt = 24sin T dT/dt
s ds/dt = -12sin T dT/dt

when T = 90°, s = 5, sinT = 1 , dT/dt = 2 rad/sec

5 ds/dt = -12(1)(2)
ds/dt = -24/5 = -4.8 inches/sec

at that precise instant, the third side is decreasing at 4.8 inches/sec

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